1. Field of the Invention
The invention is related to the field of Nuclear Magnetic Resonance (NMR) apparatus and methods. In particular, the invention is directed towards the use of regression techniques for analysis of NMR data and for determination of the properties of materials being examined using a NMR apparatus.
2. Description of the Related Art
The description of the invention and its background are approached in the context of measurement while drilling apparatus and methods for analysis of properties of earth formation. It is to be understood that the invention is not limited to this field of study.
NMR methods are among the most useful non-destructive techniques of material analysis. When hydrogen nuclei are placed in an applied static magnetic field, a small majority of spins are aligned with the applied field in the lower energy state, since the lower energy state in more stable than the higher energy state. The individual spins precess about the applied static magnetic field at a resonance frequency also termed as Larmor frequency. This frequency is characteristic to a particular nucleus and proportional to the applied static magnetic field. An alternating magnetic field at the resonance frequency in the Radio Frequency (RF) range, applied by a transmitting antenna to a subject or specimen in the static magnetic field flips nuclear spins from the lower energy state to the higher energy state. When the alternating field is turned off, the nuclei return to the equilibrium state with emission of energy at the same frequency as that of the stimulating alternating magnetic field. This RF energy generates an oscillating voltage in a receiver antenna whose amplitude and rate of decay depend on the physicochemical properties of the material being examined. The applied RF field is designed to perturb the thermal equilibrium of the magnetized nuclear spins, and the time dependence of the emitted energy is determine by the manner in which this system of spins return to equilibrium magnetization. The return is characterized by two parameters: T1, the longitudinal or spin-lattice relaxation time; and T2, the transverse or spin-spin relaxation time.
Measurements of NMR parameters of fluid filling the pore spaces of the earth formations such as relaxation times of the hydrogen spins, diffusion coefficient and/or the hydrogen density is the bases for NMR well logging. NMR well logging instruments can be used for determining properties of earth formations including the fractional volume of pore space and the fractional volume of mobile fluid filling the pore spaces of the earth formations.
Various sequences (selectable length and duration) of RF magnetic fields are imparted to the material, which are being investigated to momentarily re-orient the nuclear magnetic spins of the hydrogen nuclei. RF signals are generated by the hydrogen nuclei as they spin about their axes due to precession of the spin axes. The amplitude, duration and spatial distribution of these RF signals are related to properties of the material which are being investigated by the particular NMR techniques being used. In the well logging environment, contrast is high between free and bound fluids based on their relaxation times, between oil and water based on their relaxation times and diffusion coefficient. Based on NMR measurements, it is possible to infer something about the porosity distribution of earth formations and the fluids therein.
Methods of using NMR measurements for determining the fractional volume of pore space and the fractional volume of mobile fluid are described, for example, in Spin Echo Magnetic Resonance Logging: Porosity and Free Fluid Index Determination, M. N. Miller et al, Society of Petroleum Engineers paper no. 20561, Richardson, Tex., 1990. In porous media there is a significant difference in T1 and T2 relaxation time spectrum of fluids mixture filling the pore space. Thus, for example, light hydrocarbons and gas may have T1 relaxation time of about several seconds, while T2 may be thousand times less. This phenomenon is due to diffusion effect in internal and external static magnetic field gradients. Internal magnetic field gradients are due to magnetic susceptibility difference between rock formation matrix and pore filling fluid.
Since oil is found in porous rock formation, the relationships between porous rocks and the fluids filling their pore spaces are extremely complicated and difficult to model. Nuclear magnetic resonance is sensitive to main petrophysical parameters, but has no capabilities to establish these complex relationships. Oil and water are generally found together in reservoir rocks. Since most reservoir rocks are hydrophilic, droplets of oil sit in the center of pores and are unaffected by the pore surface. The water-oil interface normally does not affects relaxation, therefore, the relaxation rate of oil is primarily proportional to its viscosity. However, such oil by itself is a very complex mixture of hydrocarbons that may be viewed as a broad spectrum of relaxation times. In a simplest case of pure fluid in a single pore there are two diffusion regimes that govern the relaxation rate. Rocks normally have a very broad distribution of pore sizes and fluid properties. Thus it is not surprising that magnetization decays of fluid in rock formations are non-exponential. The most commonly used method of analyzing relaxation data is to calculate a spectrum of relaxation times. The Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence is used to determine the transverse magnetization decay. The non-exponential magnetization decays are fit to the multi-exponential form:
                              M          ⁡                      (            t            )                          =                              ∑                          i              =              1                        L                    ⁢                                          ⁢                                    m              ⁡                              (                                  T                                      2                    ⁢                    i                                                  )                                      ⁢                          ⅇ                                                -                  t                                /                                  T                                      2                    ⁢                    i                                                                                                          (        1        )            where M(t) represents the spin echo amplitudes, equally spaced in time, and the T2i are predetermined time constants, equally spaced on a logarithm scale, typically between 0.25 ms and 4000 ms. The set of m are found using a regularized nonlinear least squares technique. The function m(T2i), conventionally called a T2 distribution, usually maps linearly to a volumetrically weighted distribution of pore sizes.
The calibration of this mapping is addressed in several publications. Prior art solutions seek a solution to the problem of mathematical modeling of the received echo signals by the use of several techniques, including the use of non-linear regression analysis of the measurement signal; non-linear least square fit routines, as disclosed in U.S. Pat. No. 5,023,551 to Kleinberg et al, and others. Other prior art techniques include a variety of signal modeling techniques, such as polynomial rooting, singular value decomposition (SVD) and miscellaneous refinements thereof, to obtain a better approximation of the received signal. A problem with prior art signal compressions is that some information is lost.
U.S. Pat. No. 4,973,111 to Haacke describes a method for parametric image reconstruction from sampled NMR measurement data. In the method disclosed therein, the desired object function is approximated by a series of known model functions having a finite number of unknown parameters. Because the direct equations are highly non-linear, the problem is simplified by using all-pole parameter estimation, in which the unknown parameters are the roots of a polynomial equation. The coefficients of this equation are obtained as the solution vector to a system of linear prediction equations, which involve the received measurement data. The solution of the linear prediction system, as suggested in Haacke, is found by applying Singular Value Decomposition (SVD) to the linear prediction data matrix of the measurement signal. This approach is shown to reduce the effects of the measurement noise and estimate the order of the model functions.
Due to the large size of the involved matrices, however, the method of Haacke is computationally quite intensive and while suitable for off-line processing does not lend itself to real-time applications of NMR well logging. In addition, the method does not take into account information about the material under investigation or the measurement process, which can be used to simplify the computations.
U.S. Pat. No. 5,363,041 to Sezginer teaches use of a SVD and compression of raw NMR data and further a non-negative linear least square fit to obtain a distribution function. U.S. Pat. No. 5,517,115 to Prammer discloses a method of using a priori information about the nature of the expected signals to obtain an approximation of the signal using a set of pre-selected basis functions. A singular value decomposition is applied to a matrix incorporating information about the basis functions, and is stored off-line in a memory. During the actual measurement, the apparatus estimates a parameter related to the SNR of the received NMR echo trains and uses it to determine a signal approximation model in conjunction with the SVD of the basis function matrix.
All of the above discussed prior art methods rely on a two step procedure. For example, in the first step, the NMR data are inverted to give a distribution of relaxation times (T1 or T2). In the second step, some inference is drawn about the formation fluids and porosity distribution based on the relaxation time distribution. There is a certain amount of empiricism involved in each of the steps of the two step procedure, resulting in possible accumulation of errors from the individual steps. The two step procedure is avoided in U.S. Pat. No. 6,040,696 to Ramakrishnan et al. wherein an inversion method is used to derive parameters of the pore distribution in carbonates.
Thus, notwithstanding the advances in the prior art, it is perceived that the problems involved in the parameter model estimation used in NMR sensing methods for well logging have not yet been resolved. No efficient solutions have been proposed to combine advanced mathematical models with simple signal processing algorithms to increase the accuracy and numerical stability of the parameter estimates. Existing solutions require the use of significant computational power which makes the practical use of those methods inefficient, and frequently impossible to implement in real-time applications.